Just to add a little to the fire: When you go to the airport do they check your ID before you get on the plane or after?
IMHO implementing a 2FA or a ID verification wouldnt be a burden on anyone but the bumbling bureaucrats. This problem isnt limited to UI, its a major problem (10's of billions) at the IRS. Who will send a refund before they even check the return or the income.
They send the refund to the first filer, with no verification! But when it comes to collecting the funds, they will hunt you down if your EIN doesnt match your entity name on the most obscure forms...